Question

Rationalize the denominator and simplify ${\displaystyle \frac{5\sqrt{2}-\sqrt{3}}{3\sqrt{2}-\sqrt{5}}}$

Answer 1

Hint:
In this question first we will find the rationalizing factor of the denominator and then we will multiply and divide the rationalizing factor with the expression given in the question and then we will just simplify the expression by using basic rules of simplification.

Formula used:
We know that $a^2-b^2=\left(a-b\right)\left(a+b\right)$ . The other formula used in this question is $\left(a+b\right)\left(c+d\right)=ac+ad+bc+bd$

Complete step by step solution:
The given expression is ${\displaystyle \frac{5\sqrt{2}-\sqrt{3}}{3\sqrt{2}-\sqrt{5}}}$ . The denominator of the expression is $3\sqrt{2}-\sqrt{5}$ therefore, its rationalizing factor will be $3\sqrt{2}+\sqrt{5}$ . Now, multiply and divide the rationalizing factor with ${\displaystyle \frac{5\sqrt{2}-\sqrt{3}}{3\sqrt{2}-\sqrt{5}}}$ . Therefore, we can write
$\Rightarrow {\displaystyle \frac{5\sqrt{2}-\sqrt{3}}{3\sqrt{2}-\sqrt{5}}}\times {\displaystyle \frac{3\sqrt{2}+\sqrt{5}}{3\sqrt{2}+\sqrt{5}}}$
Now, the above expression can be written as follows:
$\Rightarrow {\displaystyle \frac{\left(5\sqrt{2}-\sqrt{3}\right)\left(3\sqrt{2}+\sqrt{5}\right)}{\left(3\sqrt{2}-\sqrt{5}\right)\left(3\sqrt{2}+\sqrt{5}\right)}}$
We know that $a^2-b^2=\left(a-b\right)\left(a+b\right)$. Now, use this formula in the denominator of the above expression and use the formula $\left(a+b\right)\left(c+d\right)=ac+ad+bc+bd$ in the numerator of the above expression . Therefore, we can write
$\Rightarrow {\displaystyle \frac{\left(5\sqrt{2}-\sqrt{3}\right)\left(3\sqrt{2}+\sqrt{5}\right)}{{\left(3\sqrt{2}\right)}^2-{\left(\sqrt{5}\right)}^2}}$
Now, simplify the above expression. Therefore, the above expression can be written as:
$$\begin{gathered} \Rightarrow {\displaystyle \frac{30+5\sqrt{10}-3\sqrt{6}-\sqrt{15}}{18-5}} \\ \Rightarrow {\displaystyle \frac{30+5\sqrt{10}-3\sqrt{6}-\sqrt{15}}{13}} \end{gathered}$$

Hence, in this question we have rationalize the denominator and also simplified the expression given in the question.

Additional information:
While solving this type of question on simplification just try to make the denominator of the expression which is to be simplified in normal form by using rationalizing factor.

Note:
In this question we need to have the knowledge of rationalizing factor because by using rationalizing factor we can rationalize the denominator and it will also help in simplifying the expression given in the question. So just be careful when you simplify the expression. We also need to remember the formula used in this question.